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Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

• $K_v/\mathbf{Q}_v$ is unramified for almost all $v$
• the degree $(K_v : \mathbf{Q}_v)$ is uniformly bounded
• add more as needed

It’s necessary to add more conditions. Do there exist conditions such that the answer to this question is “yes”?

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

• $K_v/\mathbf{Q}_v$ is unramified for almost all $v$
• the degree $(K_v : \mathbf{Q}_v)$ is uniformly bounded
• add more as needed

It’s necessary to add more conditions. Do there exist conditions such that the answer to this question is “yes”?

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

• $K_v/\mathbf{Q}_v$ is unramified for almost all $v$.
• add more as needed

It’s necessary to add more conditions. Do there exist conditions such that the answer to this question is “yes”?

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

• $K_v/\mathbf{Q}_v$ is unramified for almost all $v$
• the degree $(K_v : \mathbf{Q}_v)$ is uniformly bounded
• add more as needed

It’s necessary to add more conditions. Do there exist conditions such that the answer to this question is “yes”?

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

• $K_v/\mathbf{Q}_v$ is unramified for almost all $v$.
• the set $\Omega := \{v\in\Sigma \mid K_v = \mathbf{Q}_v\}$ has positive lower density

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$, a “$v$-adic field”, even when $v$ is archimedean.

Let $(K_v)_{v\in\Sigma}$ be a collection of $v$-adic fields such that:

• $K_v/\mathbf{Q}_v$ is unramified for almost all $v$.
• add more as needed

It’s necessary to add more conditions. Do there exist conditions such that the answer to this question is “yes”?

Does there exist a number field $K/\mathbf{Q}$ such that each $K_v$ is the completion of $K$ at some place lying over $v$?